Improved Bonferroni Inequalities via Abstract Tubes: Inequalities and Identities of Inclusion-Exclusion Type

✍ Scribed by Klaus Dohmen (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2003
• Tongue: English
• Leaves: 122
• Series: Lecture Notes in Mathematics 1826
• Edition: 1
• Category: Library

✦ Synopsis

This introduction to the recent theory of abstract tubes describes the framework for establishing improved inclusion-exclusion identities and Bonferroni inequalities, which are provably at least as sharp as their classical counterparts while involving fewer terms. All necessary definitions from graph theory, lattice theory and topology are provided. The role of closure and kernel operators is emphasized, and examples are provided throughout to demonstrate the applicability of this new theory. Applications are given to system and network reliability, reliability covering problems and chromatic graph theory. Topics also covered include Zeilberger's abstract lace expansion, matroid polynomials and Möbius functions.

✦ Table of Contents

1 Introduction and Overview....Pages 1-4
2 Preliminaries....Pages 5-8
3 Bonferroni Inequalities via Abstract Tubes....Pages 9-18
4 Abstract Tubes via Closure and Kernel Operators....Pages 19-43
5 Recursive Schemes....Pages 44-46
6 Reliability Applications....Pages 47-81
7 Combinatorial Applications and Related Topics....Pages 82-99
Bibliography....Pages 100-109

✦ Subjects

Combinatorics; Order, Lattices, Ordered Algebraic Structures; Probability Theory and Stochastic Processes